Cheapest way is to take a cab to Puerto Plata and take a bus from there to Santiago.

The very very cheapest way would be to take a motoconcho from the airport to the highway to the bus stop Parada de autobuses and take a guaua to Santiago and another on tho Puerto Plata.

The most comfy buses are CaribeTours, and Metro and perhaps several others. Smaller 18 passenger buses are about the same price, more crowded and not too comfy. Their advantage is that they run very frequently.
Google Autobus Santiago Puerto Plata and you will find times and rates.

Cab to the bus station in Santiago. Then Bus to Puerto Plata. Caribe Tours or Metro are your two best options for buses.
It does matter when you arrive, since the buses do not run late. If you arrive at night or early in the AM, plan to spend the night unless you want to risk a taxi ride at night.